The Long-Range Interactions

Again we consider the simplest case of gaussian equivalent links. The interaction (6.1), or (2.1), implies that the thermal Green s function for the entire chain is 

In order to pass to this limit, it is noted that IF,// is the interaction energy between two equivalent bonds per unit length squared. The interaction energy for two equivalent bonds of length As and As is then approximately Wfj ASf AsJP. Again, as in (3.12), this approximation becomes exact in the limit (6.7). Thus for the discrete chain the end-to-end vector Green s function is 

Upon passage to the continuous limit (6.7) becomes the Green s function for the continuous chain,  

We note that (6.12) exhibits translational invariance, due to the isotropy of space, as it must. Thus, if in (6.12) the transformation 

The relationship between the Wiener integral (3.20) and the simple diffusion equation (3.21) suggests that it might be instructive to convert (6.12) to a differential equation. As also noted by Whittington, for the case of a discrete chain (6.12) can be expressed only in terms of the solution of a hierarchy of integro-differential equations. The derivation in the continuous case is presented in Appendix B for convenience, although the result is quoted here. Define the three-point Green s function as 

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The long-range interactions between a pair of molecules are detemiined by electric multipole moments and polarizabilities of the individual molecules. MuJtipoJe moments are measures that describe the non-sphericity of the charge distribution of a molecule. The zeroth-order moment is the total charge of the molecule Q = Yfi- where q- is the charge of particle and the sum is over all electrons and nuclei in tlie molecule. The first-order moment is the dipole moment vector with Cartesian components given by... 

The situation for electrolyte solutions is more complex theory confimis the limiting expressions (originally from Debye-Htickel theory), but, because of the long-range interactions, the resulting equations are non-analytic rather than simple power series.) It is evident that electrolyte solutions are ideally dilute only at extremely low concentrations. Further details about these activity coefficients will be found in other articles. 

The long-range interaction V(R) between two atomic/molecular species can be decomposed into... 

Sutton and Chen extended the potential to longer range to enable the study of certain problems such as the interactions between clusters of afoms [Sutton and Chen 1990]. Their objective was to combine the superior Fiimis-Sinclair description of short-range interactions with a van der Waals tail to model the long-range interactions. The form of the Sutton-Chen potential is ... 

Pc- (c) Dipole density p. (d) Water contribution to the surface potential x calculated from the charge density Pc by means of Eq. (1). All data are taken from a 150 ps simulation of 252 water molecules between two mercury phases with (111) surface structure using Ewald summation in two dimensions for the long-range interactions. 

Siirprinsingly, despite very different diffu.se intensity maps, we find nearly concentration independent interactions (figure. 3). The similarity between the two sets of interactions is unexpected since there is an important difference of concentration between the two samples, and the experiments have been done at different temperatures. Meanwhile, this proves that the long ranged interactions [V, ...,Vq) are significant. In particular, Vg (which corresponds to the vector ( 0), in units of the fee cube) is relatively high and stable for both sets. 

The natural orbitals %2v and %3p are, in contrast to the hydrogenlike functions, localized within approximately the same region around the nucleus as the Is orbital. This means that the polarization caused by the long-range interaction is associated mainly with an angular deformation of the electronic cloud on each atom. If %2p and %3p are expanded in the standard hydrogen-like functions, an appreciable contribution will again come from the continuum. 

Theta temperature is one of the most important thermodynamic parameters of polymer solutions. At theta temperature, the long-range interactions vanish, segmental interactions become more effective and the polymer chains assume their unperturbed dimensions. It can be determined by light scattering and osmotic pressure measurements. These techniques are based on the fact that the second virial coefficient, A2, becomes zero at the theta conditions. 

If the long-range interaction 17(1, 3) can be neglected compared with 1/(1, 2), the (direct) triplet correlations can be written as ... 

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